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\title{Introduction to Quantum Mechanics\\Lecture 1}
\author{Brian Greene\footnote{These lecture notes were TeX'd by Alexander Berenbeim and Carlos Hernandez}} % ADD YOUR NAME HERE IF YOU WORK ON THIS LECTURE's NOTES
\date{September 12, 2011}

\maketitle

\section{The Weirdness of the World}

The image of the world given to us by classical mechanics can be traced directly to Newton's Second Law: 
\begin{equation}
\label{eq:newton}
\vec{F}=m\cfrac{\partial^2x}{\partial t^2} 
\end{equation}
The second law implies that the evolution of the world is \emph{fully} determined by the force of motion and initial conditions. For centuries, this image was  corroborated by empirical observation, and even today, it withstands scrutiny within certain bounds. However, two great developments in the last 100 years have come to show that at incredibly small and incredibly large scales, that this is not the case. Quantum mechanics and relativity both have shown us that the world is not Newton's fixed system, but something altogether stranger. In this course, we will be focusing on Quantum mechanics, and show that Quantum Mechanics $\sim$ Classical Mechanics, that QM is an approximation of deeper physical laws.


First, we will develop the mathematical formalism of QM. The formalism must describe experimental data, and it will reduce to Classical Mechanics in appropriate limiting conditions. The experiments leading to Quantum Mechanics began with experiments in black-body radiation, Einstein's paper on the photo-electric effect, atomic spectra, and ultimately, the double-slit experiment.

\section{The Double-Slit Experiment}
Although this is a simplified account of the actual experiment, it should be acceptable to describe the phenomenon. Scientists used an electron gun firing at a block of Nickel crystal. One day the evacuated chamber exploded. The scientists conducting the experiment heated up the Nickel crystal to clean it off. When they repeated the experiment, they witnessed that the electrons displayed a new pattern of behavior. This is because the structure of the crystal was altered.


What was observed is analogous to a double slit experiment, seen in Figure 2, and violated the classical mechanics assumption of Figure 1.

\begin{figure}[h!]
\centering
\href{http://www.columbia.edu/~cxh2102/qm/Double_slit_experiment_with_marbles.swf}{Click for demonstration}
\caption{The Classical Mechanical assumption of the double slit is distributed in this manner}
\end{figure}
\begin{figure}[h!]
\centering
\href{http://www.columbia.edu/~cxh2102/qm/Single_electrons_build_up_interference_pattern.swf}{Click for demonstration}
\caption{The interference pattern that was actually observed. Courtesy of Hitachi, Ltd.}
\end{figure}
The observed distribution raises several serious questions:
\begin{itemize}
\item Does the "wave" of electrons behave like $H_2O$ waves? Do individual electrons exhibit wavelike motion?
\item
If an electron is related to a wave- what is that wave? What is 'waving'? What does the wave represent?
\item
If electrons, and particles in general, are associated with a wave, what is the wave equation?
\item
How does the equation morph into Newton's equation?
\end{itemize}

\section{Schr\"odinger's Equation}
Before we actually can describe what is going on in the differential equation describing the time evolution of the wave:
\begin{equation}
\label{eq:schrodinger}
i\hbar\frac{\partial\psi(x,t)}{\partial t}= \frac{-\hbar^2}{2m}\frac{\partial^2\psi(x,t)}{\partial x^2}+V(x)\psi(x,t)
\end{equation}
it behooves us to first recognize that the behavior of the function $\psi(x,t)$, which describes the behavior of the electron.


Initially, there was the proposal that an electron was spread out, rather than some point-like particle. However, the notion of an electron smearing out via \emph{wave density} is wrong. Electrons are not like parts of a wave that scatter the electron by spreading it out.


Max Born 'solved' this conundrum by describing the wave observed in figure 2 as a \emph{probability wave}. The probability wave described where the particle was expected to be spatially (and as we shall see, also describes the expected velocity of the particle). The bigger the amplitude of a region of the wave, the larger the probability that the particle would be within that bounded region. Consequently, Classical Mechanics is verboten at the quantum level since we cannot look at electrons as if they follow a fixed trajectory described classically as $x(t)$. Instead, we can think of electrons are part of a dynamical probability wave.



\section{Waves As Probabilities}

And now you must be asking yourself, but how can waves be probabilities? Aren't there intervals of the wave that take on negative values? And what about integrating the wave along $\mathbb{R}^n$, and getting a value that isn't 1? Well there are all sorts of cool mathematical and algebraic consequences of using the wave equation to describe an electron's behavior that will address this.

\subsection{The Math}
First, we don't look at $\psi(x,t)$ when computing the probability of an electron's location. Rather, what we want is:
\begin{equation} 
\int\limits_{-\infty}^{\infty} |\psi(x,t)|^2\,dx=1
\end{equation}
In the cases where taking the integral over $\mathbb{R}^n$ gives us a value that is not equal to 1, as long as the value is not $\infty$, we can \emph{normalize} the $\psi(x,t)$. What is great about taking the norm of $\psi(x,t)$ is that waves can take on negative and complex values, which from an algebraic standpoint, means we have at our disposal the well-understood \emph{field} of complex numbers. Of course, this is from a mathematician's perspective. What about the physics?

\subsection{The Physics}
Recall that in classical mechanics, conservative forces come from potential. Regarding experimental data, we can measure both $x(t)$ and $\dot{x}(0)$. The result: we can determine the measured object's \emph{trajectory}, which can be described by Eq.~(\ref{eq:newton}).


In QM and Schr\"odinger's equation, Eq.~(\ref{eq:schrodinger}), V is the same potential as before, but note the $2^{nd}$ order derivative! As a result, we cannot determine any subsequent trajectory with certainty. Indeed, as we will see later, we will actually run into an inequality halting what we can know, known as Heisenberg's Uncertainty Principle. Bonus points if you can see it in the equation right now.
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